Noncommutative Algebra and Applications
Projeto Temático FAPESP No.2015/09162-9, coordenado por César Polcino Milies
Research group seminars
2019 seminars:
Lecturer |
Title |
Date |
Time |
Room |
Pedro Russo (IME-USP) |
On the existence of free noncyclic groups in finite dimensional division rings with involution |
November 05, 2019 |
14:30-15:30 |
242-A |
Abstract: Lichtman has conjectured that a non central normal subgroup N of the multiplicative group of a division ring D contains a non cyclic free subgroup. We address the particular case where D is finite dimensional over its center k, which is supposed to be uncountable and whose characteristic is different from 2, and is endowed with a k-involution. Under these circumstances, if N contains the nonzero members of k, then it contains a free non cyclic subgroup whose free generators are symmetric. |
Lecturer |
Title |
Date |
Time |
Room |
Jairo Z. Gonçalves (IME-USP) |
Constructing free and free symmetric pairs in cyclic algebras with an involution |
October 29, 2019 |
14:30-15:30 |
242-A |
Abstract: Let A = (K/F, \sigma, Y, b) be a cyclic algebra, where K/F is a Galois extension over the non absolute field F with Galois group < \sigma> of order n, and let b \in F. Let us assume, moreover, that A=(K/F, \sigma, Y, b) satisfies the relations Y^n = b and YaY^{-1}=a^{\sigma} , for all a \in K. Let U=U(A) be the group of units of A. We show how to construct pairs (u, v) in U such that <u, v> is a free group of rank two. Also, if char F \neq 2, * is an F-involution of A such that K*=K, and if A is not the quaternion algebra with the simplectic involution, then we exhibit a pair (u, v) in U(A) such that u* = u, v* = v and <u, v> is a free group of rank two. |
Lecturer |
Title |
Date |
Time |
Room |
Tran Giang Nam (Institute of Mathematics, VAST, Vietnam) |
Realizing corners of Leavitt path algebras as Steinberg algebras, with corresponding connections to graph C*-algebras (part 2) |
October 22, 2019 |
14:30-15:30 |
242-A |
Abstract: We show that the endomorphism ring of any nonzero finitely generated projective module over the Leavitt path algebra L_K(E) of an arbitrary graph E with coefficients in a field K is isomorphic to a Steinberg algebra. This yields in particular that every nonzero corner of the Leavitt path algebra of an arbitrary graph is isomorphic to a Steinberg algebra. This in its turn gives that every K-algebra with local units which is Morita equivalent to the Leavitt path algebra of a row-countable graph is isomorphic to a Steinberg algebra. Moreover, we prove that a corner by a projection of a C*-algebra of a countable graph is isomorphic to the C*-algebra of an ample groupoid. (Joint work with Gene Abrams and Misha Dokuchaev.) |
Lecturer |
Title |
Date |
Time |
Room |
Tran Giang Nam (Institute of Mathematics, VAST, Vietnam) |
Realizing corners of Leavitt path algebras as Steinberg algebras, with corresponding connections to graph C*-algebras |
October 08, 2019 |
14:30-15:30 |
242-A |
Abstract: We show that the endomorphism ring of any nonzero finitely generated projective module over the Leavitt path algebra L_K(E) of an arbitrary graph E with coefficients in a field K is isomorphic to a Steinberg algebra. This yields in particular that every nonzero corner of the Leavitt path algebra of an arbitrary graph is isomorphic to a Steinberg algebra. This in its turn gives that every K-algebra with local units which is Morita equivalent to the Leavitt path algebra of a row-countable graph is isomorphic to a Steinberg algebra. Moreover, we prove that a corner by a projection of a C*-algebra of a countable graph is isomorphic to the C*-algebra of an ample groupoid. (Joint work with Gene Abrams and Misha Dokuchaev.) |
Lecturer |
Title |
Date |
Time |
Room |
André Pereira (UFRRJ) |
Algebras of p-groups of class 2 over the rationals |
October 01, 2019 |
14:30-15:30 |
242-A |
Abstract: The goal of this presentation is to analyze p-groups, M and N, with nilpotency class 2 and same rational group algebras. We prove that if QM = QN, then their commutator subgroups are equal and the centers, Z(M) and Z(N), have their orders preserved. After this, we apply the result to Frattini Central p- group, this is, the Frattini subgroup is in the center of the group. To finish, we present an example of two p-groups of order p^7, with nilpotency class 2 such that they have the same rational group algebra, but the centers of these groups are not isomorphic. |
Lecturer |
Title |
Date |
Time |
Room |
Raul Ferraz (IME-USP) |
One weight codes in some classes of group rings |
September 17, 2019 |
14:30-15:30 |
242-A |
Abstract: Let F_q be a finite field with q elements and G be a group of order n. Firstly we give conditions to ensure that a cyclic code in F_q G is a one-weight code in the semi simple case (that is when gcd(n,q) = 1). Further we consider the case when G is abelian, and also when gcd(n,q) >1. (Joint work with Ruth Nascimento Ferreira from Universidade Tecnológica Federal do Paraná, Guarapuava-PR.) |
Lecturer |
Title |
Date |
Time |
Room |
Ángel del Río (University of Murcia, Spain) |
The Zassenhaus Conjecture for cyclic-by-abelian groups, with some proofs |
September 10, 2019 |
14:30-15:30 |
242-A |
Abstract: Let G be a finite group. The Zassenhaus conjecture states that every torsion unit of augmentation 1 in the integral group algebra is conjugate in the rational group algebra to an element of G. Although a counterexample has been found recently in the class of metabelian groups, it has been proved for same large classes of groups including nilpotent groups and cyclic-by-abelian groups, and it is open for the class of supersolvable groups. We will present some positive results for the class of cyclic-by-nilpotent groups obtained recently in cooperation with Mauricio Caicedo. |
Lecturer |
Title |
Date |
Time |
Room |
Mayumi Makuta (IME-USP) |
Obstructions to extensions of semilattices of groups by groups (part 2) |
August, 20, 2019 |
14:30-15:30 |
242-A |
Abstract: In this talk, we give an interpretation of the third partial cohomology group (defined by Dokuchaev and Khrypchenko) with values in an abelian semilattice of groups in terms of extensions of a semilattice of groups by a group. For thisend, we define a partial abstract kernel inspired on the classic case of group cohomology and the case of inverse semigroup cohomology. |
Lecturer |
Title |
Date |
Time |
Room |
Mayumi Makuta (IME-USP) |
Obstructions to extensions of semilattices of groups by groups |
August, 13, 2019 |
14:30-15:30 |
242-A |
Abstract: In this talk, we give an interpretation of the third partial cohomology group (defined by Dokuchaev and Khrypchenko) with values in an abelian semilattice of groups in terms of extensions of a semilattice of groups by a group. For thisend, we define a partial abstract kernel inspired on the classic case of group cohomology and the case of inverse semigroup cohomology. |
Lecturer |
Title |
Date |
Time |
Room |
Jairo Z. Gonçalves (IME-USP) |
Free pairs of symmetric elements in normal subgroups of division rings (part 2) |
June 11, 2019 |
14:30-15:30 |
242-A |
Abstract: Let D be a division ring with central subfield k of characteristic different from 2, let * be a k-involution of D, and let N be a normal subgroup of the multiplicative group of D. We show that if G \subseteq N is a *-stable nonabelian subgroup that is either torsion-free polycyclic-by-finite but not abelian-by-finite, or finite of odd order, then N contains a pair (u, v) of elements such that u^*=u, v^*=v and such that \langle u, v \rangle is a noncyclic free group. One aspect of the above proof requires that we extend a theorem of Bergman on invariant ideals in commutative group algebras. This new result is surely of interest in its own right. It appears in the Appendix and can be read independently of the remainder of the paper. |
Lecturer |
Title |
Date |
Time |
Room |
Jairo Z. Gonçalves (IME-USP) |
Free pairs of symmetric elements in normal subgroups of division rings |
June 04, 2019 |
14:30-15:30 |
242-A |
Abstract: Let D be a division ring with central subfield k of characteristic different from 2, let * be a k-involution of D, and let N be a normal subgroup of the multiplicative group of D. We show that if G \subseteq N is a *-stable nonabelian subgroup that is either torsion-free polycyclic-by-finite but not abelian-by-finite, or finite of odd order, then N contains a pair (u, v) of elements such that u^*=u, v^*=v and such that \langle u, v \rangle is a noncyclic free group. One aspect of the above proof requires that we extend a theorem of Bergman on invariant ideals in commutative group algebras. This new result is surely of interest in its own right. It appears in the Appendix and can be read independently of the remainder of the paper. |
Lecturer |
Title |
Date |
Time |
Room |
Mikhailo Dokuchaev (IME-USP) |
A Chase-Harrison-Rosenberg sequence for a partial Galois extension of commutative rings (part 3) |
May 28, 2019 |
14:30-15:30 |
242-A |
Abstract: In
a paper published in 1965 S.U. Chase, D.K. Harrison and A.
Rosenberg obtained a series of results on Galois
extensions of commutative rings, including several
equivalent definitions, a Galois correspondence and a
seven term exact sequence, involving Galois cohomology
groups, Picard groups and the relative Brauer group. The
sequence is seen as a common generalization of two
classical facts on Galois extensions of fields: the
Hilbert's Theorem 90 and the isomorphism of the relative
Brauer group with the second Galois cohomology group. In a
joint paper with M. Ferrero and A. Paques (2007) we
defined Galois extensions in the context of partial group
actions and obtained a Galois correspondence. It is
natural to extend the Chase-Harrison-Rosenberg sequence
for the case of partial Galois extensions of commutative
rings. In a joint paper with A. Paques and H. Pinedo we
built up homomorphisms, which are analogues of those in
the Chase-Harrison-Rosenberg sequence. Now in a preprint
with A. Paques, H. Pinedo and I. Rocha we prove that the
sequence is exact. We shall recall the |
Lecturer |
Title |
Date |
Time |
Room |
Mikhailo Dokuchaev (IME-USP) |
A Chase-Harrison-Rosenberg sequence for a partial Galois extension of commutative rings (part 2) |
May 21, 2019 |
14:30-15:30 |
242-A |
Abstract: In
a paper published in 1965 S.U. Chase, D.K. Harrison and A.
Rosenberg obtained a series of results on Galois
extensions of commutative rings, including several
equivalent definitions, a Galois correspondence and a
seven term exact sequence, involving Galois cohomology
groups, Picard groups and the relative Brauer group. The
sequence is seen as a common generalization of two
classical facts on Galois extensions of fields: the
Hilbert's Theorem 90 and the isomorphism of the relative
Brauer group with the second Galois cohomology group. In a
joint paper with M. Ferrero and A. Paques (2007) we
defined Galois extensions in the context of partial group
actions and obtained a Galois correspondence. It is
natural to extend the Chase-Harrison-Rosenberg sequence
for the case of partial Galois extensions of commutative
rings. In a joint paper with A. Paques and H. Pinedo we
built up homomorphisms, which are analogues of those in
the Chase-Harrison-Rosenberg sequence. Now in a preprint
with A. Paques, H. Pinedo and I. Rocha we prove that the
sequence is exact. We shall recall the |
Lecturer |
Title |
Date |
Time |
Room |
Mikhailo Dokuchaev (IME-USP) |
A Chase-Harrison-Rosenberg sequence for a partial Galois extension of commutative rings |
May 07, 2019 |
14:30-15:30 |
242-A |
Abstract: In
a paper published in 1965 S.U. Chase, D.K. Harrison and A.
Rosenberg obtained a series of results on Galois
extensions of commutative rings, including several
equivalent definitions, a Galois correspondence and a
seven term exact sequence, involving Galois cohomology
groups, Picard groups and the relative Brauer group. The
sequence is seen as a common generalization of two
classical facts on Galois extensions of fields: the
Hilbert's Theorem 90 and the isomorphism of the relative
Brauer group with the second Galois cohomology group. In a
joint paper with M. Ferrero and A. Paques (2007) we
defined Galois extensions in the context of partial group
actions and obtained a Galois correspondence. It is
natural to extend the Chase-Harrison-Rosenberg sequence
for the case of partial Galois extensions of commutative
rings. In a joint paper with A. Paques and H. Pinedo we
built up homomorphisms, which are analogues of those in
the Chase-Harrison-Rosenberg sequence. Now in a preprint
with A. Paques, H. Pinedo and I. Rocha we prove that the
sequence is exact. We shall recall the |
Lecturer |
Title |
Date |
Time |
Room |
Tran Giang Nam (Institute of Mathematics, VAST, Vietnam) |
Leavitt
path algebras - Something for everyone: |
April 30, 2019 |
14:30-15:30 |
242-A |
Abstract: The
rings studied by students in most first-year algebra
courses have the "Invariant Basis Number'' property:
for every pair of positive integers m and |
Lecturer |
Title |
Date |
Time |
Room |
Tran Giang Nam (Institute of Mathematics, VAST, Vietnam) |
Leavitt
path algebras - Something for everyone: |
April 23, 2019 |
14:30-15:30 |
242-A |
Abstract: The
rings studied by students in most first-year algebra
courses have the "Invariant Basis Number'' property:
for every pair of positive integers m and |
Lecturer |
Title |
Date |
Time |
Room |
Daniel Kawai (IME-USP) |
Graded division rings |
April 9, 2019 |
14:30-15:30 |
242-A |
Abstract: In
ring theory, we often study properties about ring
epimorphisms from a given ring R to division rings, that
we call epic R-division rings. Paul Cohn discovered a
correspondence between epic R-division rings and some sets
of square matrices that he called prime matrix ideals.
Moreover, he obtained a criterion for the existence of an
epic R-division ring, and a criterion for R to be
embeddable in a division ring. Given a ring R, Cohn
considered a kind of morphism of R-division rings that he
called specialization, and the existence of a
specialization was shown to be equivalent to the property
that every matrix over R that becomes invertible in L
becomes invertible in K, and it was shown equivalent to
the property that every rational relation over R that is
valid in K is valid in L. Therefore, he also studied some
important class of rings R there is an initial epic
R-division ring, such as Sylvester domains, and
pseudo-Sylvester domains, firs and semifirs. Moreover, if
G is a free group and K is a division ring, the group ring
K[G] is a semifir. |
Lecturer |
Title |
Date |
Time |
Room |
Antonio Giambruno (Universitá di Palermo, Italia) |
Polynomial identities and star-fundamental algebras |
March 26, 2019 |
14:30-15:30 |
242-A |
Abstract: In
the theory of polynomial identities in characteristic zero
developed by Kemer an important role is played by the
so-called fundamental algebras. These are finite
dimensional algebras defined in terms of multialternating
polynomials and one of the main features is that any
finite dimensional algebra has the same identities as a
finite direct sum of fundamental algebras. In the last
years we have developed together with La Mattina e Polcino
Milies a theory of fundamental |
Lecturer |
Title |
Date |
Time |
Room |
Makar Plakhotnyk (IME-USP) |
Combinatorial automorphisms of quasi semi metrics |
March 19, 2019 |
14:30-15:30 |
242-A |
Abstract: Quasi
semi metrics on a finite set can be represented by square
n\times n real non-negative matrices A = (\alpha_{pq})
such that \alpha_{ii}=0 for all i and \alpha_{ij} +
\alpha_{jk}\geq \alpha_{ik} for all i, j, k. If we assume
that the entries \alpha_{ij} are integers we obtain the
definition of a non-negative |
Lecturer |
Title |
Date |
Time |
Room |
Cesar Polcino Milies (IME-USP) |
Essential idempotents in group algebras and coding theory |
March 12, 2019 |
14:30-15:30 |
242-A |
Abstract: A primitive idempotent e in a semisimple group algebra F_qG is called essential if e\widehat{H} = 0 for every normal subgroup H of G. We give some criteria to decide when a primitive idempotent is essential, and compute the number of essential idempotents in cyclic group algebras. Then, we find a relation among essential idempotents in different group algebras. Along the way, we explore their significance for coding theory. |
Lecturer |
Title |
Date |
Time |
Room |
Itailma Rocha (UFCG) |
Uma
sequência
exata
relacionada a uma extensão
de |
February 26, 2019 |
14:30-15:30 |
249-A |
Abstract: Para uma extensão de Galois de anéis comutativos, Chase-Harrison-Rosenberg construíram uma sequência exata de sete termos que envolve o grupo de Picard, o grupo de Brauer relativo e grupos de cohomologias. Essa sequência é vista como uma generalização de dois fatos importantes da teoria galoisiana de corpos, a saber, o Teorema 90 de Hilbert e o isomorfismo de grupo de Brauer relativo com o segundo grupo de cohomologia. A sequência foi generalizada por Miyashita para o contexto de anéis não comutativos com unidade. Mais tarde, El Kaoutit e Gómez-Torrencillas generalizaram o resultado de Miyashita para uma extensão de anéis não comutativos e não unitais, apenas com um conjunto de unidades locais. A sequência de Chase-Harrison-Rosenberg também foi considerada para ações parciais por Dokuchaev, Paques e Pinedo, que construíram uma versão para uma extensão de Galois parcial de anéis comutativos. Nesta tese, elaboramos uma versão da sequência no contexto de ações parciais para uma extensão de anéis não comutativos com unidade. A sequência apresentada aqui generaliza a sequência dada por Miyashita. |